# The Jacobian of $(x,y)\mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.

I am given the map $$(x,y)\mapsto (x+y^2,y+x^2)$$. I am unable to find the Jacobian by making the substitution $$u=x+y^2$$ and $$v=y+x^2$$. Any hints would be appreciated.

(I am trying to find whether the map is area preserving? I know "the map $$f:\mathbb{R}^n\to\mathbb{R}^n$$ is area and orientation preserving iff the determinant of the Jacobian is $$\pm1$$".)

• What is the definition of Jacobian? – zoidberg Dec 9 '18 at 2:39

The Jacobian of a multivariable function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$$ such that $$f(x,y)=(u(x,y),v(x,y))$$ is:
$$\textbf{J}(f(x,y)) = det\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{pmatrix}$$
where $$det$$ denotes the determinant of the matrix.
Calculating the partial derivatives of $$u(x,y)=x+y^2$$ and $$v(x,y)=y+x^2$$ we have that the Jacobian of your given function is:
$$det\begin{pmatrix} 1 & 2y \\ 2x & 1 \\ \end{pmatrix}=1-4xy$$